%   ORIGINAL: h4/powser/DIFFS__NEG
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/real/REAL__NEG__RMUL: !y x. h4/realax/real__neg (h4/realax/real__mul x y) = h4/realax/real__mul x (h4/realax/real__neg y)
% Assm: h4/powser/diffs0: !c. h4/powser/diffs c = (\n. h4/realax/real__mul (h4/real/real__of__num (h4/num/SUC n)) (c (h4/num/SUC n)))
% Goal: !c. h4/powser/diffs (\n. h4/realax/real__neg (c n)) = (\n. h4/realax/real__neg (h4/powser/diffs c n))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_reals_REALu_u_NEGu_u_RMUL]: !y x. h4/realax/real__neg (h4/realax/real__mul x y) = h4/realax/real__mul x (h4/realax/real__neg y)
% Assm [h4s_powsers_diffs0]: !c x. h4/powser/diffs c x = h4/realax/real__mul (h4/real/real__of__num (h4/num/SUC x)) (happ c (h4/num/SUC x))
% Goal: !_0. (!c n. happ (happ _0 c) n = h4/realax/real__neg (happ c n)) ==> (!c x. h4/powser/diffs (happ _0 c) x = h4/realax/real__neg (h4/powser/diffs c x))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q281848,TV_Q281844]: ![V_f, V_g]: (![V_x]: s(TV_Q281844,happ(s(t_fun(TV_Q281848,TV_Q281844),V_f),s(TV_Q281848,V_x))) = s(TV_Q281844,happ(s(t_fun(TV_Q281848,TV_Q281844),V_g),s(TV_Q281848,V_x))) => s(t_fun(TV_Q281848,TV_Q281844),V_f) = s(t_fun(TV_Q281848,TV_Q281844),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_reals_REALu_u_NEGu_u_RMUL, axiom, ![V_y, V_x]: s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,V_y))))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_x),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,V_y)))))).
fof(ah4s_powsers_diffs0, axiom, ![V_c, V_x]: s(t_h4s_realaxs_real,h4s_powsers_diffs(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_c),s(t_h4s_nums_num,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_x))))),s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_c),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_x)))))))).
fof(ch4s_powsers_DIFFSu_u_NEG, conjecture, ![V_uu_0]: (![V_c, V_n]: s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_c))),s(t_h4s_nums_num,V_n))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,happ(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_c),s(t_h4s_nums_num,V_n))))) => ![V_c, V_x]: s(t_h4s_realaxs_real,h4s_powsers_diffs(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),t_fun(t_h4s_nums_num,t_h4s_realaxs_real)),V_uu_0),s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_c))),s(t_h4s_nums_num,V_x))) = s(t_h4s_realaxs_real,h4s_realaxs_realu_u_neg(s(t_h4s_realaxs_real,h4s_powsers_diffs(s(t_fun(t_h4s_nums_num,t_h4s_realaxs_real),V_c),s(t_h4s_nums_num,V_x))))))).
